What is the limit as x approaches infinity of ((x+1)/(x+3))^{x/5} ?

The limit as x approaches infinity of ((x+1)/(x+3))^{x/5} is 1/(e^{2/5)}

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limit as x approaches infinity of ((x+1)/(x+3))^{x/5}

x \to \infty lim ((\frac{x + 1}{x + 3})^{\frac{x}{5}})

\frac{1}{e ^{\frac{2}{5}}}

Decimal Notation

0.67032 \dots

Solution steps

x \to \infty lim ((\frac{x + 1}{x + 3})^{\frac{x}{5}})

Apply exponent rules

= x \to \infty lim (e^{\frac{x}{5} l n (\frac{x + 1}{x + 3})})

Apply the Limit Chain Rule:

\frac{1}{e ^{\frac{2}{5}}}

= \frac{1}{e ^{\frac{2}{5}}}

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What is the limit as x approaches infinity of ((x+1)/(x+3))^{x/5} ?
The limit as x approaches infinity of ((x+1)/(x+3))^{x/5} is 1/(e^{2/5)}